Some generalizations of Hermite–Hadamard type inequalities

Some generalizations and refinements of Hermite–Hadamard type inequalities related to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta$$\end{document}η-convex functions are investigated. Also applications for trapezoid and mid-point type inequalities are given.


Introduction and preliminaries
This paper generalizes some well-known results for Hermite-Hadamard integral inequality by generalizing the convex function factor of the integrand to be an η-convex function. The obtained results have as particular cases those previously obtained for convex functions in the integrand.
Let I be an interval in real line R. Consider η : A × A → B for appropriate A, B ⊆ R.
In fact above definition geometrically says that if a function is η-convex on I, then its graph between any x, y ∈ I is on or under the path starting from y, f y and ending at (x, f y + η f (x), f y . If f (x) should be the end point of the path for every x, y ∈ I, then we have η x, y = x − y and the function reduces to a convex one. There exists η-convex functions for some bifunctions η that are not convex. We have the following simple examples: Example 2 (Gordji et al. 2015) a. Consider a function f : R → R defined by and define a bifunction η as η x, y = −x − y, for all x, y ∈ R − = (−∞, 0]. It is not hard to check that f is a η-convex function but not a convex one.
b. Define the function f : R + → R + by and define the bifunction η : Then f is η-convex but is not convex.
The following theorem is an important result: Theorem 3 (Gordji et al. 2016) Suppose that f : I → R is a η-convex function and η is bounded from above on f (I) × f (I). Then f satisfies a Lipschitz condition on any closed interval [a, b] contained in the interior I • of I. Hence, f is absolutely continuous on [a, b] and continuous on I • .
Remark 4 As a consequence of Theorem 3, an η-convex function f : The following simple lemma is required.
Proof Assertions are consequence of this fact:

□
We have a basic lemma: Lemma 6 Let f : [a, b] → R be a η-convex function. Then for any t ∈ [0, 1] we have the inequalities and Proof If in (2) we put t instead of 1 − t and then add that inequality with (2) we have: If in (7) we replace a with b and add the result with (7), then we have (3). Now, if in (2) we put a instead of b and then add that inequality with (2) we get: for all t ∈ [0, 1], which is equivalent to (4). If we change a with b, and t with 1 − t in (2) and then add that inequality with (2) we get: for all t ∈ [0, 1] and the inequality (5) is proved.
Finally since we have and then by using (2) we can obtain (6) □

Hermite-Hadamard type inequalities
In this section we obtain some Hermite-Hadamard type integral inequalities which improve right and left side of (1) respectively.
Theorem 1 and All of inequalities (11)-(13) are different views for right side of generalized Hermite-Hadamard inequalities and finally can be stated as a unique form of If we suppose that η x, y = x − y, then we recapture right side of (1). Also we can obtain the following result:

]). Then we have the inequalities:
Proof From (6) we have for any t ∈ [0, 1]. Integrating over t we get the first inequality in (15). Now Using properties of Lemma 5 along with integrating rules gives (15) , then by Theorem 3 we have which gives a refinement for left side of (1). If we suppose that η x, y = x − y, then we recapture left side of (1).

Trapezoid and mid-point type inequalities
An interesting question in (1), is estimating the difference between left and middle terms and between right and middle terms. In this section we investigate about this question, when the absolute value of the derivative of a function is η-convex. We need Lemma 2.1 in Kirmaci (2004): (16)

Remark 2
In Lemma 1, if we use the change of variable x = tb + (1 − t)a, then Using Lemma 1, we can prove the following theorem to estimate the difference between the middle and left terms in (1).
Theorem 3 Suppose that f : [a, b] → R is a differentiable mapping and f ′ is an η-convex mapping on [a, b] with a bounded η from above. Then where Proof From η-convexity of f ′ , Theorem 3 and Lemma 1 it follows that On the other hand according to Remark 2 we have Then we can deduce the result from